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It is possible to go by train and it generally takes 3.5 to 4 hours and is via London and undergound between St Pancras and Liverpool Street. It may be viable if you take away costs of the car journey and parking from the cost of the rail fare.
It seems to be very difficult to find public transport to Harwich. Parking is available alongside the ship for £5 per day according to this....http://www.celebritycruises.com/beforeyourcruise/faq/home.do;jsessionid=0000SWlNXbGLVYS3TKAv2NmJ9mU:12hdbcveb?faqSubjectName=Directions&faqId=447&pagename=faq_answers